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Exponential Functions

You have seen the procedure by which a new function, the exponential function $q(t) = q_0 \exp(kt)$, was constructed from a power series just to provide a solution to the differential equation $\dot{q} = k \, q$. (There are, of course, other ways of ``inventing'' this delightful function, but I like my story.) You may suspect that this sort of procedure will take place again and again, as we seek compact notation for the functions that ``solve'' other important differential equations. Indeed it does! We have Legendre polynomials, various Bessel functions, spherical harmonics and many other ``named functions'' for just this purpose. But - pleasant surprise! - we can get by with just the ones we have so far for almost all of Newtonian Mechanics, provided we allow just one more little ``extension'' of the exponential function . . . .